A word to begin: the literature has a number of sophisticated approaches to the logic of statements about probability (notably Fagin, Halpern, and Megiddo, Inform and Compute 87 (1990): 78-128). What I want to do here is much less ambitious and more elementary. But I hope that by keeping our focus very narrow, on just what I will call elementary statements and their ‘internal’ logic (rather than their Boolean combinations) we can get some interesting insights into probabilistic thinking.
That the probability of rain today is 60% is surely the paradigm example of a proposition that assigns a probability. Its form, using the usual symbols, is P(A) = x, and I will say that this statement is satisfied by any probability measure which assigns value x to proposition (or event) A.
I will P(A) = x as my paradigm example of what should count as an elementary statement. Then I hope to arrive at a useful concept of elementary statements in general, with a clear notion of when they are satisfied by a probability measure. We’ll soon see that P(A) < x, P(A) ≥ x, and the like, are other good candidates for this status. Given all that it will be possible to define the semantic correlate of logical consequence:
if Q, R are elementary statements then Q entails R exactly if every probability measure which satisfies Q also satisfies R.
The task will be first to settle on a useful concept of elementary statement, and then to explore the variety of such statements. These will surely include ones like P(A) ≥ 0.5, and other inequalities, and perhaps some combinations of those.
As my main clue I will take the concept of convexity. That is a notion that appears in many places, wherever it is useful to use the word “between”. For example, a geometric figure is convex if, for any two points that lie inside, and the points between those lie inside as well. So a solid cube or sphere is convex, but a squiggly worm is not.
A set of numbers is convex if for any two numbers in it, the numbers between them belong to it as well, in the following precise sense:
the number x is a convex combination of numbers y and z if and only there is a number a ε [0,1] such that x = ay + (1-a)z
So an interval is convex (contains all convex combinations of its own members), but the set of prime numbers is not.
Of course the two uses of “between” don’t stand for the same relationship — in each case, to talk about convexity we have to fix what will correspond to familiar cases of betweenness.
Here is the notion for probabilities. If P(A) = x is satisfied by both p and p’ then it is also satisfied by an mixture (convex combination) of p and p’, and that means by any function defined by an equation of this form:
p” = ap + (1-a)p’, with a ε [0,1]
Spelled out that means that
p”(A) = ap(A) + (1-a)p'(A), for all propositions A in the relevant domain.
The really important point to remember: as we can readily verify, if p” is a convex combination of p and p’ then p”(A) is always a number between p(A) and p'(A), inclusive.
The precise definition capturing the relevant notion of ‘between’ has the same form for probability functions as it has for numbers. And similarly, a set of probability measures is convex if and only if it contains all the mixtures of its members.
Still informally ( before we put it all in terms of models and model structures) let us refer to the set of probability measures that satisfy a statement Q as |Q|, and call it the semantic value of Q. Then we note that |P(A) = x| is convex. Trivially so, because the only number between x and x, inclusive, is x itself. But |P(A) < 0.5| is also convex, for if two numbers are less than 0.5 then so is every number that lies between them. Similarly for |P(A) ≥ 0.5| and the like.
This feature, that the semantic value is a convex set, I will choose as defining mark of a useful concept of elementary statement.
Of course there is a connection between convexity for the numbers assigned as probabilities and convexity for probability. Note that each of the examples can be written as P(A) ε I, for an (open, closed, or half-open) interval. Thus P(A) < 0.5 is the same as P(A) ε [0,0.5).
Theorem. If I is any set of numbers then |P(A) ε I| is convex if I is convex.
Suppose first that I is convex and that p, p’ satisfy the statement P(A) ε I. Then p(A) and p'(A) are in I, and since I is convex, any number between them is in I also. But if p” is a convex combination of p and p’ then p”(A) is a number between those two, so is also in I. Hence p” satisfies P(A) ε I.
Theorem. If |P(A) ε I|is convex then there is a convex subset J of I such that |P(A) εI| = |P(A) ε J|.
Suppose that |P(A) ε I| is empty. The empty set Λ is convex by definition, definitely a subset of I, and clearly |P(A) ε I| = |P(A) ε Λ|.
If |P(A) ε I|is not empty, let J be the set of numbers x such that P(A) = x and x ε I. Then if |P(A) ε I|is convex, hence contains all convex combinations of its members, J will contain all convex combinations of its members. Thus |P(A) ε I| = |P(A) ε J|
Conclusion: We have a general notion of elementary statement, namely one whose semantic value is a convex set of probability measures. And we have one general form of statements which are elementary, namely |P(A) ε I|, with I a convex set of numbers (more specifically, a convex subset of [0,1]).
Question: are there other forms of that elementary statements can have? What about combinations, like conjunction, disjunction, negation?
This will be the topic of the next post. In the Appendix below I will make things precise, as we have them so far. Mostly, though, I will proceed in the informal way of the above.
APPENDIX: making it precise
Probability is a modality, so it seems apt to set this up in the same was as we do for normal modal logics, in so far as that is possible.
Syntax: a set of proposition terms A, B, C, … The sentences will be of form P(A) R x, where R is any relation any of the relations of equality or inequality, <, ≤, >, etc.. and P(A) ε I.
Semantics: A frame (‘sample space’) is a couple K = <W, F>, where W is a non-empty set and F a field of subsets of W, called the propositions. We can think of the members of W as possible worlds or as events of some sort. The family of probability measures with domain F will be called P(M).
A model structure is a triple M = <W, F, I> where <W, F> is a frame an I is an interpretation, that is, I assigns members of F to the proposition terms. The set of probability measures on F will still be called P(M). A member p of P(M) satisfies P(A) R x relative to I if and only if p(A) bears R to x.
NOTE: I am using the capital letters from early in the alphabet equally for proposition terms and for the propositions for which they stand. If needed to be more precise, the last phrase in the preceding paragraph would be “if and only if p(I(A)) bears R to x.
Theorem. If R is any relation of equality or inequality, the set of probability measures in P(M) that satisfy P(A) R x is convex. Similarly for P(A) ε I: see theorems above for details.
Notation. If Q is a statement and M is a model structure then the set of elements of P(M) that satisfy Q will be called|M, Q|. In a context where a single model structure M is under discussion this will be abbreviated to |Q|.
Definition. A statement Q is elementary if and only if |M, Q| is convex, for every model structure M.
All the statements in our syntax so far are therefore elementary, by our definition.
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